3.1.1 What do the most expensive houses look like?

From slicing the data, it looks like the most expensive houses are very well constructed, have tens of thousands of square feet of property, and have 5 or more bedrooms.

3.1.2 What is the average price for each condition level?

From the slice below, average prices seem to trend upward along with condition; average prices are in the hundreds of thousands.

##   condition  price
## 1         1 341067
## 2         2 328149
## 3         3 542089
## 4         4 521274
## 5         5 612402

3.1.3 What is the average price for each grade level?

From the slice below, we can see that price generally trends upward along with grade.

##    grade   price
## 1      3  262000
## 2      4  212002
## 3      5  248524
## 4      6  301920
## 5      7  402566
## 6      8  542944
## 7      9  773513
## 8     10 1071771
## 9     11 1496842
## 10    12 2201285
## 11    13 3709615

3.2.1 What do the largest houses look like?

Going into this project, we hypothesized that larger houses would be priced higher than smaller houses. House size is determined in large part by “sqft_living”, of which “bathrooms” and “bedrooms” are a part.

It is apparent here that the largest houses are also among the most expensive – they are all priced in the millions of dollars, which are outliers when compared to the dataset as a whole.

The properties with the largest amount of land are also priced highly, but not as highly as those listed in the “sqft_living” readout. This could suggest a lower correlation between housing price and sqft_lot than that between housing price and sqft_living.

4.1.1 Comparison of sqft living with price

From the scatterplot, it’s apparent that there is a relatively strong, positive correlation between housing price and living space (.70192, to be exact). That is, as living space increases, so does housing price. Note that a majority of the data points lie below 6,000 sqft, and below $2 million.

Now let’s take a look at the same data with a boxplot; this time, we have “sqft_living” categorized by 5 intervals. From this visualization as well, it’s apparent that “sqft_living” correlates positively with housing price. The last interval (10,891-13,540 sqft) seems to buck this trend, but it worth noting that only 2 houses are part of this group – a small-n population, which could explain the discrepancy.

Next, we perform the BP Test to determine the homoscedasticity of the different groups to assess if we can perfom an ANOVA test for the differences in mean values between different houses’ living space.

H0: The variances of prices are the same across the different sqft_living levels.
H1: The variances of prices are different across the different sqft_living levels.

Since the p-value is 0, which is lower than 0.05, we need to reject the null hypothesis. The variances for different sqft_living levels are different. Thus, the ANOVA test is not applicable.

4.1.2 Comparison of sqft lot with price

Next up in our exploratory data analysis is housing price vs. “sqft_lot”. How does land area correlate with housing price? According to our scatterplot, not very highly – there is a positive correlation of only .08988. This seems to suggest that sqft_lot is more weakly related to housing price than sqft_living. Indeed, the vast majority of data points in the scatterplot seem to trend upward in price with relatively small increases in land area.

Next, let’s take a look at the same data in a boxplot. Unfortunately, the visualization isn’t very readable; let’s convert housing price through a logarithmic function to improve our y-axis scale.

The modified boxplot below (with the logarithmic scale) is much easier to interpret. We can see that housing price increases as land area increases, but only to an extent. Note that houses in the 991,000-1.3M sqft and 1.3M-1.65M sqft ranges appear to buck the trend. Once again, this can be explained by the fact that only a few houses are part of these two intervals – only 4 to be exact.

Thus, we perform the BP Test to determine the homoscedasticity of the different groups to assess if we can perfom an ANOVA test for the differences in mean values between different houses’ living space.

BP test:
H0: The variances of prices are the same across the different sqft_lot levels.
H1: The variances of prices are different across the different sqft_lot levels.

ANOVA test:
H0: There are no differences between the mean prices of the different sqft.lot levels.
H1: The mean prices of the different sqft.lot levels are different.

## 
##  studentized Breusch-Pagan test
## 
## data:  kc_house_data$price ~ sqft.lot
## BP = 0.2, df = 4, p-value = 1

##                Df           Sum Sq      Mean Sq F value    Pr(>F)    
## sqft.lot        4    3896930121642 974232530411    7.24 0.0000081 ***
## Residuals   21591 2906956970577216 134637440164                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = kc_house_data$price ~ sqft.lot)
## 
## $sqft.lot
##                         diff      lwr     upr p adj
## 330K-660K-520-330K    130987   -15181  277155 0.104
## 660K-991K-520-330K    619510   265542  973477 0.000
## 991K-1.3M-520-330K    -10511  -588471  567449 1.000
## 1.3M-1.65M-520-330K   160322  -840687 1161331 0.992
## 660K-991K-330K-660K   488523   105684  871361 0.005
## 991K-1.3M-330K-660K  -141498  -737577  454580 0.967
## 1.3M-1.65M-330K-660K   29335  -982244 1040914 1.000
## 991K-1.3M-660K-991K  -630021 -1307691   47650 0.083
## 1.3M-1.65M-660K-991K -459187 -1520893  602518 0.763
## 1.3M-1.65M-991K-1.3M  170833  -985006 1326672 0.994

Note that the p-value of BP test is 0.996, which is greater than 0.05. Thus, the variances are the same for different groups. Then we use the ANOVA test to analyze the differences of mean prices. Since the p-value is 0.000008033, which is lower than 0.05, we need to reject the null hypothesis. The mean prices of the different sqft.lot levels are different. From the Turkey test, we can see that the p-values for the following pairs are the lowest: the 660k-991k and 520k-330k, 660k-991k and 330K-660K, 660k-991k and 991K-1.3M. And the p-value for other pairs are very high, which means the mean price of 660k-991k(the median level) are higher than others’. Thus, the land area of 660k-991k are the most popular level for houses in WA.

4.1.3 Comparison of number of bedrooms with price

Here, we have a logarithmic box plot of housing price vs. “bedrooms”. There appears to be a clear trend: as the number of bedrooms increases, housing price increases as well. The 9-11 interval bucks the trend slightly, but again, this can be explained by the fact that only 10 houses are part of this interval, compared with 21586 total for the others.

Next, we perform the BP Test to determine the homoscedasticity of the different groups to assess if we can perfom an ANOVA test for the differences in mean values between different houses’ living space.

H0: Price variance is the same across bedroom groups.
H1: Price variance is different across bedroom groups.

Since the p-value is 0, which is lower than 0.05, we need to reject the null hypothesis. The variances for different bedroom groups are different. Thus, the ANOVA test is not applicable.

Since the anova test is not applicable, we want to use chi-square test to see the correlation between the number of bedrooms and house price. The prices and number of bathrooms are divided into different groups by the cut function to make it categorical, so we can perform the chi-square test on them.

H0: The price and number of bedrooms are independent.
H1: The price and number of bedrooms are not independent.

## 
##  Pearson's Chi-squared test
## 
## data:  bed_p
## X-squared = 2287, df = 21, p-value <0.0000000000000002

## 
##  Pearson's Chi-squared test
## 
## data:  bath_p
## X-squared = 4534, df = 21, p-value <0.0000000000000002

Since both p-values are lower than 0.05, we need to reject the null hypothesis. Thus, the number of bedrooms and the number of bathrooms are not independent from the price of house. They are correlated. Generally, the price increases when the number of bedrooms and bathrooms become greater.

4.2.1 Comparison of condition with price

Here, we compare “condition” with housing price. Once again, “condition” represents an index from 1 to 5, with the lowest number representing poor condition. Once we take a look at the boxplot below (the second one is logarithmized for clearer visualization), it becomes clear that apartment condition correlates positively with housing price.

Then we do the BP test to see the homoscedasticity for different condition groups.

BP test:
H0: Price variance is the same accross different conditions.
H1: Price variance is different accross different conditions.

Since the p-value is 0.292, which is greater than 0.05, we failed to reject the null hypothesis. The price variance between different condition groups is the same. The ANOVA test is applicable.

Next we do the ANOVA test to see the mean prices of houses with different condition.

ANOVA test:
H0: There are no differences between the mean prices of different conditions.
H1: The mean prices of the different conditions are not equal.

##                            Df           Sum Sq       Mean Sq F value
## kc_house_data$condition     4   19739857713567 4934964428392    36.9
## Residuals               21591 2891114042985296  133903665554        
##                                      Pr(>F)    
## kc_house_data$condition <0.0000000000000002 ***
## Residuals                                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = kc_house_data$price ~ kc_house_data$condition)
## 
## $`kc_house_data$condition`
##       diff     lwr    upr p adj
## 2-1 -12918 -213478 187642 1.000
## 3-1 201022   15459 386585 0.026
## 4-1 180207   -5637 366051 0.062
## 5-1 271335   84389 458280 0.001
## 3-2 213940  136914 290965 0.000
## 4-2 193125  115424 270825 0.000
## 5-2 284253  203953 364552 0.000
## 4-3 -20815  -36519  -5111 0.003
## 5-3  70313   44676  95950 0.000
## 5-4  91128   63529 118727 0.000

Since the p-value is 0.0000000000000000000000000000008494, which is lower than 0.05, we need to reject the null hypothesis. The mean prices of the different conditions are not equal. From the Tukey test we know that conditions 1 and 2 have very large p-values. Other p-values are very low. Basically, the price increases when the condition is better.

4.2.2 Comparison of grade with price

Here we have a boxplot comparing “grade” with housing price. Once again, “grade” represents an index from 1 to 13, with the lowest number representing poor construction and design. The trend is clear: construction and design grade correlate positively with housing price.

Then we do BP test to see if the ANOVA test is applicable.

H0: The variances of prices are the same across different grades.
H1: The variances of prices are not equal across different grades.

Since the p-value is 0, which is lower than 0.05, we need to reject the null hypothesis. The price variance for different grades are not equal. Thus, the ANOVA test is not applicable.

Now we want to see whether condition and grade are somehow related; for this, we must perform a chi-square test.

H0: Condition and grade of houses are independent from each other.
H1: Condition and grade of house are not independent from each other.

## 
##  Pearson's Chi-squared test
## 
## data:  cond.tbl
## X-squared = 1457, df = 40, p-value <0.0000000000000002

Since the p-value is 0, which is lower than 0.05, we need to reject the null hypothesis. Thus, condition and grade are not independent. They are correlated. Generally, the grade of construction is better when house conditions are better.

4.3.1 Comparison of year they were built with price

Here, we have a comparison of “yr_built” with housing price. Once we take a look at the logarithmic boxplot, we see no obvious trends. Housing price trends downward from 1900-1969, and then picks back up from 1970-2015. What might explain this? Well, “yr_built” does not take “yr_renovated” into account. For instance, two equivalent houses built in the same year could have different house prices, depending on if one has been renovated while the other hasn’t.

Let’s construct the same boxplots, but this time indexed by “yr_renovated”.

Unfortunately, the vast majority of properties in this dataset have never been renovated (20682 to be exact). This means that only 914 properties have been renovated. This makes the resulting boxplots somewhat uninformative – the larger population boxplot (not renovated) largely mirrors the patterns of the previous graph, and the smaller population graph (renovated) is based on a population too small to run meaningful analysis on. We have included the graphs here to showcase our thought process, but we are well aware of their limitations.

Thus, we want to get more insights into the significance of the differences we may observe throughout all these graphs and we then attempt an ANOVA test for renovated houses aggregated by yr_built.

We perform the BP Test to determine the homoscedasticity of the different groups to assess if we can perfom an ANOVA test for different periods of construction, which results in a high p-value allowing us to conduct the ANOVA for differences in price among older and newer renovated houses.

H0: There are no differences between the mean prices of the different yr_built renovated houses.
H1: The mean prices of the different yr_built renovated houses are different.

##              Df          Sum Sq      Mean Sq F value Pr(>F)  
## yr_built      4   3524693302886 881173325722     2.4  0.048 *
## Residuals   909 333684139527418 367089262406                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = price ~ yr_built, data = yr_R)
## 
## $yr_built
##                       diff     lwr     upr p adj
## 1924-1946-1900-1923 -27818 -172093  116458 0.985
## 1947-1969-1900-1923  54244  -85395  193882 0.826
## 1970-1992-1900-1923 184015  -26330  394360 0.119
## 1993-2015-1900-1923 280784 -466793 1028361 0.843
## 1947-1969-1924-1946  82061  -57719  221841 0.495
## 1970-1992-1924-1946 211832    1394  422271 0.048
## 1993-2015-1924-1946 308601 -439002 1056205 0.792
## 1970-1992-1947-1969 129771  -77516  337059 0.428
## 1993-2015-1947-1969 226540 -520182  973263 0.922
## 1993-2015-1970-1992  96769 -666344  859881 0.997

Since the p-value is 0.04853, which is less than 0.05, we need to reject the null hypothesis. Thus, the prices of the renovated are different across yr_built. From the Tukey test, we can see that only the mean price of renovated houses built in 1970-1992 and in 1924-1946 are different from each other. Although given a p-value fairly higher compared to the rest obtained in this analysis, we can also notice that with a more conservative significance level we would not find differences significant among the various years of constructions of the houses renovated, which is reasonable given the fact that most houses were renovated around the same period and thus more likely to be priced similarly.
Even though we could not perform an ANOVA test for the overall sample of houses by yr_built because of bp test results in that case, the boxplots confirm the possibility that also there would be no significant differences in price between older and newer houses, regardless their renovation status provided in this sample.

4.3.2 Comparison of year renovated with price

Here, we’ve graphed housing price by yr_renovated itself. This graph also showcases only 914 properties – the ones that were renovated. Generally speaking, as “yr_renovated” approaches the present day, price increases. The exception is between the 1924-1946 and 1947-1969 intervals; note however, that only 9 properties occupy the first interval.

5.1.1 LSRL Model building

Below is our regression model, along with a comprehensive correlation plot.

## 
## Call:
## lm(formula = price ~ . - sqft_lot - yr_built, data = h2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1706400  -142615   -21697   101727  4133425 
## 
## Coefficients: (1 not defined because of singularities)
##                Estimate Std. Error t value             Pr(>|t|)    
## (Intercept)    80549.20    7851.17   10.26 < 0.0000000000000002 ***
## bedrooms      -64528.79    2446.40  -26.38 < 0.0000000000000002 ***
## bathrooms       5499.45    3852.17    1.43              0.15341    
## sqft_living      340.68       4.99   68.33 < 0.0000000000000002 ***
## floors         14875.02    4299.43    3.46              0.00054 ***
## sqft_above       -36.54       5.03   -7.26      0.0000000000004 ***
## sqft_basement        NA         NA      NA                   NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 257000 on 21590 degrees of freedom
## Multiple R-squared:  0.51,   Adjusted R-squared:  0.509 
## F-statistic: 4.49e+03 on 5 and 21590 DF,  p-value: <0.0000000000000002

## 
## Call:
## lm(formula = price ~ . - sqft_lot - yr_built - sqft_basement - 
##     bathrooms, data = h2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1711175  -142619   -21684   101736  4134675 
## 
## Coefficients:
##              Estimate Std. Error t value             Pr(>|t|)    
## (Intercept)  81162.16    7839.61   10.35 < 0.0000000000000002 ***
## bedrooms    -63928.53    2410.05  -26.53 < 0.0000000000000002 ***
## sqft_living    343.98       4.42   77.89 < 0.0000000000000002 ***
## floors       17336.10    3938.78    4.40    0.000010807578329 ***
## sqft_above     -37.41       5.00   -7.49    0.000000000000074 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 257000 on 21591 degrees of freedom
## Multiple R-squared:  0.51,   Adjusted R-squared:  0.509 
## F-statistic: 5.61e+03 on 4 and 21591 DF,  p-value: <0.0000000000000002

##    bedrooms sqft_living      floors  sqft_above 
##        1.55        5.37        1.48        5.59

##      price            bedrooms      sqft_living      sqft_above  
##  Min.   :  78000   Min.   : 1.00   Min.   :  370   Min.   : 370  
##  1st Qu.: 322000   1st Qu.: 3.00   1st Qu.: 1430   1st Qu.:1190  
##  Median : 450000   Median : 3.00   Median : 1910   Median :1560  
##  Mean   : 540198   Mean   : 3.37   Mean   : 2080   Mean   :1789  
##  3rd Qu.: 645000   3rd Qu.: 4.00   3rd Qu.: 2550   3rd Qu.:2210  
##  Max.   :7700000   Max.   :11.00   Max.   :13540   Max.   :9410  
##                                                                  
##      floors         grade      condition
##  Min.   :1.00   7      :8973   1:   29  
##  1st Qu.:1.00   8      :6065   2:  170  
##  Median :1.50   9      :2615   3:14020  
##  Mean   :1.49   6      :2038   4: 5677  
##  3rd Qu.:2.00   10     :1134   5: 1700  
##  Max.   :3.50   11     : 399            
##                 (Other): 372

## 
## Call:
## lm(formula = price ~ ., data = h3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1571904  -122108   -22410    88448  4645573 
## 
## Coefficients:
##               Estimate Std. Error t value             Pr(>|t|)    
## (Intercept)  119945.02  235012.20    0.51              0.60979    
## bedrooms     -27981.05    2255.58  -12.41 < 0.0000000000000002 ***
## sqft_living     229.90       4.37   52.66 < 0.0000000000000002 ***
## sqft_above      -89.75       4.65  -19.29 < 0.0000000000000002 ***
## floors        25574.69    3798.02    6.73       0.000000000017 ***
## grade4        52746.97  235247.78    0.22              0.82259    
## grade5        47680.72  231471.19    0.21              0.83680    
## grade6        77263.78  231054.85    0.33              0.73808    
## grade7       110396.50  231036.91    0.48              0.63278    
## grade8       183631.54  231071.92    0.79              0.42680    
## grade9       327826.56  231145.29    1.42              0.15613    
## grade10      530666.16  231258.19    2.29              0.02176 *  
## grade11      829031.75  231551.46    3.58              0.00034 ***
## grade12     1357863.56  232717.34    5.83       0.000000005462 ***
## grade13     2552196.47  240535.18   10.61 < 0.0000000000000002 ***
## condition2   -60791.74   46520.27   -1.31              0.19130    
## condition3   -62199.90   43243.71   -1.44              0.15035    
## condition4    -5587.43   43281.73   -0.13              0.89728    
## condition5    71583.51   43532.93    1.64              0.10012    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 231000 on 21577 degrees of freedom
## Multiple R-squared:  0.605,  Adjusted R-squared:  0.604 
## F-statistic: 1.83e+03 on 18 and 21577 DF,  p-value: <0.0000000000000002

##    bedrooms sqft_living  sqft_above      floors      grade4      grade5 
##        1.68        6.51        6.01        1.70       27.99      240.44 
##      grade6      grade7      grade8      grade9     grade10     grade11 
##     1847.93     5250.37     4367.68     2302.97     1077.66      393.79 
##     grade12     grade13  condition2  condition3  condition4  condition5 
##       90.02       14.10        6.85      172.49      147.02       55.66

## 
## Call:
## lm(formula = price ~ . - condition, data = h3)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1594879  -122654   -26548    89616  4612348 
## 
## Coefficients:
##               Estimate Std. Error t value             Pr(>|t|)    
## (Intercept)  203437.68  234053.01    0.87               0.3848    
## bedrooms     -25531.89    2283.29  -11.18 < 0.0000000000000002 ***
## sqft_living     241.42       4.40   54.90 < 0.0000000000000002 ***
## sqft_above     -101.49       4.69  -21.65 < 0.0000000000000002 ***
## floors        11331.97    3765.10    3.01               0.0026 ** 
## grade4       -58531.64  238316.99   -0.25               0.8060    
## grade5       -47771.75  234518.50   -0.20               0.8386    
## grade6       -24710.22  234099.87   -0.11               0.9159    
## grade7         2597.97  234076.33    0.01               0.9911    
## grade8        71562.72  234108.48    0.31               0.7598    
## grade9       212286.01  234180.94    0.91               0.3647    
## grade10      412631.72  234294.46    1.76               0.0782 .  
## grade11      707325.79  234587.29    3.02               0.0026 ** 
## grade12     1233712.59  235767.03    5.23           0.00000017 ***
## grade13     2416245.65  243679.96    9.92 < 0.0000000000000002 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 234000 on 21581 degrees of freedom
## Multiple R-squared:  0.594,  Adjusted R-squared:  0.594 
## F-statistic: 2.26e+03 on 14 and 21581 DF,  p-value: <0.0000000000000002

##    bedrooms sqft_living  sqft_above      floors      grade4      grade5 
##        1.68        6.43        5.94        1.63       27.97      240.31 
##      grade6      grade7      grade8      grade9     grade10     grade11 
##     1846.94     5247.31     4365.02     2301.52     1076.98      393.53 
##     grade12     grade13 
##       89.96       14.09

## 
## Call:
## lm(formula = price ~ . + bedrooms:sqft_living + bedrooms:floors + 
##     bedrooms:sqft_above + sqft_living:floors + sqft_living:sqft_above + 
##     floors:sqft_above, data = h4)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -3695612  -119737   -25450    86085  3346184 
## 
## Coefficients:
##                             Estimate    Std. Error t value
## (Intercept)             245214.63848  229404.26957    1.07
## bedrooms                -51836.57174    7294.93923   -7.11
## sqft_living                 86.13585      16.53152    5.21
## sqft_above                 -19.70424      21.95986   -0.90
## floors                   36232.45799   14886.82743    2.43
## grade4                  -45196.54696  233098.37846   -0.19
## grade5                  -15410.52348  229421.70528   -0.07
## grade6                   21127.44809  229036.61201    0.09
## grade7                   74426.60451  229048.12877    0.32
## grade8                  160884.19715  229093.63765    0.70
## grade9                  310355.78334  229162.33811    1.35
## grade10                 492220.91881  229263.64399    2.15
## grade11                 721454.84294  229542.28824    3.14
## grade12                1105961.32670  230837.39838    4.79
## grade13                1837445.26776  239796.69406    7.66
## bedrooms:sqft_living       -16.02514       3.80857   -4.21
## bedrooms:floors          25728.79001    5273.85824    4.88
## bedrooms:sqft_above         20.28226       5.16049    3.93
## sqft_living:floors         101.59363       8.63259   11.77
## sqft_living:sqft_above       0.03473       0.00195   17.79
## sqft_above:floors         -177.53444       9.78398  -18.15
##                                    Pr(>|t|)    
## (Intercept)                          0.2851    
## bedrooms                  0.000000000001233 ***
## sqft_living               0.000000190166631 ***
## sqft_above                           0.3696    
## floors                               0.0149 *  
## grade4                               0.8463    
## grade5                               0.9464    
## grade6                               0.9265    
## grade7                               0.7452    
## grade8                               0.4825    
## grade9                               0.1757    
## grade10                              0.0318 *  
## grade11                              0.0017 ** 
## grade12                   0.000001669853101 ***
## grade13                   0.000000000000019 ***
## bedrooms:sqft_living      0.000025908368174 ***
## bedrooms:floors           0.000001076292883 ***
## bedrooms:sqft_above       0.000085104651817 ***
## sqft_living:floors     < 0.0000000000000002 ***
## sqft_living:sqft_above < 0.0000000000000002 ***
## sqft_above:floors      < 0.0000000000000002 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 229000 on 21575 degrees of freedom
## Multiple R-squared:  0.612,  Adjusted R-squared:  0.611 
## F-statistic: 1.7e+03 on 20 and 21575 DF,  p-value: <0.0000000000000002

##               bedrooms            sqft_living             sqft_above 
##                   17.9                   95.0                  136.2 
##                 floors                 grade4                 grade5 
##                   26.6                   28.0                  240.4 
##                 grade6                 grade7                 grade8 
##                 1848.1                 5252.3                 4369.7 
##                 grade9                grade10                grade11 
##                 2304.0                 1078.0                  393.9 
##                grade12                grade13   bedrooms:sqft_living 
##                   90.2                   14.3                  144.4 
##        bedrooms:floors    bedrooms:sqft_above     sqft_living:floors 
##                   70.0                  194.0                  150.2 
## sqft_living:sqft_above      sqft_above:floors 
##                   32.1                  173.0

## 
## Call:
## lm(formula = price ~ . + bedrooms:sqft_above + sqft_living:sqft_above, 
##     data = h4)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -3552516  -119684   -27362    86763  3536811 
## 
## Coefficients:
##                             Estimate    Std. Error t value
## (Intercept)             301536.38692  230960.70303    1.31
## bedrooms                -37942.22923    5140.56726   -7.38
## sqft_living                174.31529       5.84404   29.83
## sqft_above                -232.12312       9.01099  -25.76
## floors                   14271.05585    3722.97985    3.83
## grade4                  -35043.35004  235079.61029   -0.15
## grade5                    9116.80210  231363.17493    0.04
## grade6                   50285.38874  230969.99192    0.22
## grade7                  107201.66962  230973.41942    0.46
## grade8                  195978.96124  231016.35221    0.85
## grade9                  342536.69129  231090.06081    1.48
## grade10                 523989.90325  231190.83681    2.27
## grade11                 751707.87220  231468.92597    3.25
## grade12                1141807.53068  232773.60628    4.91
## grade13                1933822.97431  241686.51914    8.00
## bedrooms:sqft_above         12.57344       2.48837    5.05
## sqft_living:sqft_above       0.02832       0.00169   16.75
##                                    Pr(>|t|)    
## (Intercept)                         0.19171    
## bedrooms                 0.0000000000001629 ***
## sqft_living            < 0.0000000000000002 ***
## sqft_above             < 0.0000000000000002 ***
## floors                              0.00013 ***
## grade4                              0.88150    
## grade5                              0.96857    
## grade6                              0.82765    
## grade7                              0.64256    
## grade8                              0.39626    
## grade9                              0.13828    
## grade10                             0.02343 *  
## grade11                             0.00117 ** 
## grade12                  0.0000009399814052 ***
## grade13                  0.0000000000000013 ***
## bedrooms:sqft_above      0.0000004387237711 ***
## sqft_living:sqft_above < 0.0000000000000002 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 231000 on 21579 degrees of freedom
## Multiple R-squared:  0.605,  Adjusted R-squared:  0.605 
## F-statistic: 2.07e+03 on 16 and 21579 DF,  p-value: <0.0000000000000002

##               bedrooms            sqft_living             sqft_above 
##                   8.75                  11.67                  22.55 
##                 floors                 grade4                 grade5 
##                   1.64                  27.97                 240.39 
##                 grade6                 grade7                 grade8 
##                1847.89                5251.23                4368.71 
##                 grade9                grade10                grade11 
##                2303.51                1077.80                 393.79 
##                grade12                grade13    bedrooms:sqft_above 
##                  90.13                  14.24                  44.35 
## sqft_living:sqft_above 
##                  23.68

5.1.2 Final results and approved model for prediction of price

Price = 142000 +bedrooms(-31710+10.72sqft_above)+sqft_living(170+2.943sqft_above)+sqft_above(-228.6)+floors14570+grade()

Problem: As the price histogram above is quite left-skewed, it means there are many outliers whose price is very high in the dataset. While we built the model, we did not exclude the outliers as we considered these values important. As a result, our final model is also skewed a bit. It means that for low price houses, our model may predict higher-than-normal prices, and for high price houses, our model will predict lower-than-normal prices.